Questions — Edexcel (10514 questions)

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Edexcel C34 2017 June Q14
16 marks Standard +0.8
14. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-48_506_812_219_571} \captionsetup{labelformat=empty} \caption{Figure 6}
\end{figure} Figure 6 shows a sketch of the curve \(C\) with parametric equations $$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$ Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)
  1. find the coordinates of \(P\). The line \(l\) is the normal to \(C\) at \(P\).
  2. Show that an equation of \(l\) is \(y = x + 3.5\) The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
  3. Show that the area of \(S\) is given by $$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
  4. Hence, by integration, find the exact area of \(S\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
    END
Edexcel C34 2018 June Q1
6 marks Moderate -0.8
  1. Find $$\int \frac { 2 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } } \mathrm {~d} x , \quad x > 0$$
  2. Find $$\int x \cos 2 x \mathrm {~d} x$$
Edexcel C34 2018 June Q2
7 marks Moderate -0.3
2. A curve \(C\) has parametric equations $$x = \frac { 3 } { 2 } t - 5 , \quad y = 4 - \frac { 6 } { t } \quad t \neq 0$$
  1. Find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(t = 3\), giving your answer as a fraction in its simplest form.
  2. Show that a cartesian equation of \(C\) can be expressed in the form $$y = \frac { a x + b } { x + 5 } \quad x \neq k$$ where \(a , b\) and \(k\) are integers to be found.
Edexcel C34 2018 June Q3
7 marks Standard +0.3
3. $$f ( x ) = 2 ^ { x - 1 } - 4 + 1.5 x \quad x \in \mathbb { R }$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as $$x = \frac { 1 } { 3 } \left( 8 - 2 ^ { x } \right)$$ The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 1.6\) to one decimal place.
  2. Starting with \(x _ { 0 } = 1.6\), use the iteration formula $$x _ { n + 1 } = \frac { 1 } { 3 } \left( 8 - 2 ^ { x _ { n } } \right)$$ to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
  3. By choosing a suitable interval, prove that \(\alpha = 1.633\) to 3 decimal places.
Edexcel C34 2018 June Q4
10 marks Standard +0.3
4.
  1. Find the binomial expansion of $$( 1 + p x ) ^ { - 4 } , \quad | p x | < 1$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient as simply as possible in terms of the constant \(p\). $$f ( x ) = \frac { 3 + 4 x } { ( 1 + p x ) ^ { 4 } } \quad | p x | < 1$$ where \(p\) is a positive constant. In the series expansion of \(\mathrm { f } ( x )\), the coefficient of \(x ^ { 2 }\) is twice the coefficient of \(x\).
  2. Find the value of \(p\).
  3. Hence find the coefficient of \(x ^ { 3 }\) in the series expansion of \(\mathrm { f } ( x )\), giving your answer as a simplified fraction.
Edexcel C34 2018 June Q5
12 marks Standard +0.2
  1. The functions \(f\) and \(g\) are defined by $$\begin{array} { l l } \mathrm { f } : x \rightarrow \mathrm { e } ^ { 2 x } - 5 , & x \in \mathbb { R } \\ \mathrm {~g} : x \rightarrow \ln ( 3 x - 1 ) , & x \in \mathbb { R } , x > \frac { 1 } { 3 } \end{array}$$
    1. Find \(\mathrm { f } ^ { - 1 }\) and state its domain.
    2. Find \(\mathrm { fg } ( 3 )\), giving your answer in its simplest form.
    3. (a) Sketch the graph with equation $$y = | 4 x - a |$$ where \(a\) is a positive constant. State the coordinates of each point where the graph cuts or meets the coordinate axes. Given that $$| 4 x - a | = 9 a$$ where \(a\) is a positive constant,
  2. find the possible values of $$| x - 6 a | + 3 | x |$$ giving your answers, in terms of \(a\), in their simplest form.
Edexcel C34 2018 June Q6
11 marks Standard +0.3
6.
  1. Express \(\sqrt { 5 } \cos \theta - 2 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 significant figures.
  2. Solve, for \(- \pi < \theta < \pi\), $$\sqrt { 5 } \cos \theta - 2 \sin \theta = 0.5$$ giving your answers to 3 significant figures. [Solutions based entirely on graphical or numerical methods are not acceptable.] $$\mathrm { f } ( x ) = A ( \sqrt { 5 } \cos \theta - 2 \sin \theta ) + B \quad \theta \in \mathbb { R }$$ where \(A\) and \(B\) are constants. Given that the range of f is $$- 15 \leqslant f ( x ) \leqslant 33$$
  3. find the value of \(B\) and the possible values of \(A\).
Edexcel C34 2018 June Q7
5 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-22_362_766_237_589} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a hemispherical bowl.
Water is flowing into the bowl at a constant rate of \(180 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\).
When the height of the water is \(h \mathrm {~cm}\), the volume of water \(V \mathrm {~cm} ^ { 3 }\) is given by $$V = \frac { 1 } { 3 } \pi h ^ { 2 } ( 90 - h ) , \quad 0 \leqslant h \leqslant 30$$ Find the rate of change of the height of the water, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), when \(h = 15\) Give your answer to 2 significant figures.
Edexcel C34 2018 June Q8
11 marks Standard +0.3
8. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 1 \\ - 3 \\ 2 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 6 \\ 4 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 1 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.
  1. Show that \(l _ { 1 }\) and \(l _ { 2 }\) do not meet. The point \(P\) is on \(l _ { 1 }\) where \(\lambda = 0\), and the point \(Q\) is on \(l _ { 2 }\) where \(\mu = - 1\)
  2. Find the acute angle between the line segment \(P Q\) and \(l _ { 1 }\), giving your answer in degrees to 2 decimal places.
  3. Find the shortest distance from the point \(Q\) to the line \(l _ { 1 }\), giving your answer to 3 significant figures.
Edexcel C34 2018 June Q9
8 marks Standard +0.8
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-28_533_1095_258_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Diagram not drawn to scale
  1. Find $$\int \frac { 1 } { ( 2 x - 1 ) ^ { 2 } } d x$$ Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { 12 } { ( 2 x - 1 ) } \quad 1 \leqslant x \leqslant 5$$ The finite region \(R\), shown shaded in Figure 2, is bounded by the line with equation \(x = 1\), the curve with equation \(y = \mathrm { f } ( x )\) and the line with equation \(y = \frac { 4 } { 3 }\). The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  2. Find the exact value of the volume of the solid generated, giving your answer in its simplest form.
    \section*{Leave
    k}
Edexcel C34 2018 June Q10
7 marks Standard +0.8
10. The curve \(C\) satisfies the equation $$x \mathrm { e } ^ { 5 - 2 y } - y = 0 \quad x > 0 , \quad y > 0$$ The point \(P\) with coordinates ( \(2 \mathrm { e } ^ { - 1 } , 2\) ) lies on \(C\).
The tangent to \(C\) at \(P\) cuts the \(x\)-axis at the point \(A\) and cuts the \(y\)-axis at the point \(B\).
Given that \(O\) is the origin, find the exact area of triangle \(O A B\), giving your answer in its simplest form. \includegraphics[max width=\textwidth, alt={}]{a377da06-a968-438c-bec2-ae55283dae47-35_4_21_127_2042} L
Edexcel C34 2018 June Q11
9 marks Standard +0.8
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-36_601_1140_242_402} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure}
  1. By writing \(\sec \theta\) as \(\frac { 1 } { \cos \theta }\), show that when \(x = 3 \sec \theta\), $$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = 3 \sec \theta \tan \theta$$ Figure 3 shows a sketch of part of the curve \(C\) with equation $$y = \frac { \sqrt { x ^ { 2 } - 9 } } { x } \quad x \geqslant 3$$ The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 6\)
  2. Use the substitution \(x = 3 \sec \theta\) to find the exact value of the area of \(R\). [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C34 2018 June Q12
9 marks Standard +0.8
12.
  1. Show that $$\cot x - \tan x \equiv 2 \cot 2 x , \quad x \neq 90 n ^ { \circ } , n \in \mathbb { Z }$$
  2. Hence, or otherwise, solve, for \(0 \leqslant \theta < 180 ^ { \circ }\) $$5 + \cot \left( \theta - 15 ^ { \circ } \right) - \tan \left( \theta - 15 ^ { \circ } \right) = 0$$ giving your answers to one decimal place.
    [0pt] [Solutions based entirely on graphical or numerical methods are not acceptable.]
Edexcel C34 2018 June Q13
11 marks Standard +0.3
13.
  1. Express \(\frac { 1 } { ( 4 - x ) ( 2 - x ) }\) in partial fractions. The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation
    where \(k\) is a constant.
  2. solve the differential equation and show that the solution can be written as $$x = \frac { 4 - 4 \mathrm { e } ^ { 2 k t } } { 1 - 2 \mathrm { e } ^ { 2 k t } }$$ Given that \(k = 0.1\)
  3. find the value of \(t\) when \(x = 1\), giving your answer, in seconds, to 3 significant figures. The mass, \(x\) grams, of a substance at time \(t\) seconds after a chemical reaction starts is modelled by the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = k ( 4 - x ) ( 2 - x ) , \quad t \geqslant 0,0 \leqslant x < 2$$ where \(k\) is a constant. $$\text { Given that when } t = 0 , x = 0$$
(b) solve the differential equation and show that the solution can be written as
Edexcel C34 2018 June Q14
12 marks Standard +0.3
14. Given that $$y = \frac { \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
  1. show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { A x ^ { 2 } + 12 } { x ^ { 4 } \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } \quad x > 2$$ where \(A\) is a constant to be found. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-48_593_1134_865_395} \captionsetup{labelformat=empty} \caption{Figure 4}
    \end{figure} Figure 4 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \frac { 24 \left( x ^ { 2 } - 4 \right) ^ { \frac { 1 } { 2 } } } { x ^ { 3 } } \quad x > 2$$
  2. Use your answer to part (a) to find the range of f.
  3. State a reason why f-1 does not exist.
Edexcel C34 2019 June Q1
7 marks Standard +0.3
1. $$f ( x ) = 2 x ^ { 3 } + x - 20$$
  1. Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as $$x = \sqrt [ 3 ] { a - b x }$$ where \(a\) and \(b\) are positive constants to be determined.
  2. Starting with \(x _ { 1 } = 2.1\) use the iteration formula \(x _ { n + 1 } = \sqrt [ 3 ] { a - b x _ { n } }\), with the numerical values of \(a\) and \(b\), to calculate the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 3 decimal places.
  3. Using a suitable interval, show that 2.077 is a root of the equation \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.
  4. Hence state a root, to 3 decimal places, of the equation $$2 ( x + 2 ) ^ { 3 } + x - 18 = 0$$
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Edexcel C34 2019 June Q2
7 marks Moderate -0.3
2.
  1. Find \(\int \frac { 4 x + 3 } { x } \mathrm {~d} x , \quad x > 0\)
  2. Given that \(y = 25\) at \(x = 1\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { ( 4 x + 3 ) y ^ { \frac { 1 } { 2 } } } { x } \quad x > 0 , y > 0$$ giving your answer in the form \(y = [ \mathrm { g } ( x ) ] ^ { 2 }\).
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    \includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-05_52_49_2777_1886}
Edexcel C34 2019 June Q3
6 marks Moderate -0.3
3. A curve \(C\) has parametric equations $$x = \sqrt { 3 } \tan \theta , \quad y = \sec ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ The cartesian equation of \(C\) is $$y = \mathrm { f } ( x ) , \quad 0 \leqslant x \leqslant k , \quad \text { where } k \text { is a constant }$$
  1. State the value of \(k\).
  2. Find \(\mathrm { f } ( x )\) in its simplest form.
  3. Hence, or otherwise, find the gradient of the curve at the point where \(\theta = \frac { \pi } { 6 }\)
Edexcel C34 2019 June Q4
8 marks Standard +0.3
4. The curve \(C\) has equation $$3 y \mathrm { e } ^ { - 2 x } = 4 x ^ { 2 } + y ^ { 2 } + 2$$
  1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\). The point \(P\) on \(C\) has coordinates \(( 0,2 )\).
  2. Find the equation of the normal to \(C\) at \(P\) giving your answer in the form \(y = m x + c\), where \(m\) and \(c\) are constants to be found.
    (3)
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Edexcel C34 2019 June Q5
9 marks Moderate -0.3
5. A bath is filled with hot water. The temperature, \(\theta ^ { \circ } \mathrm { C }\), of the water in the bath, \(t\) minutes after the bath has been filled, is given by $$\theta = 20 + A \mathrm { e } ^ { - k t }$$ where \(A\) and \(k\) are positive constants. Given that the temperature of the water in the bath is initially \(38 ^ { \circ } \mathrm { C }\),
  1. find the value of \(A\). The temperature of the water in the bath 16 minutes after the bath has been filled is \(24.5 ^ { \circ } \mathrm { C }\).
  2. Show that \(k = \frac { 1 } { 8 } \ln 2\) Using the values for \(k\) and \(A\),
  3. find the temperature of the water 40 minutes after the bath has been filled, giving your answer to 3 significant figures.
  4. Explain why the temperature of the water in the bath cannot fall to \(19 ^ { \circ } \mathrm { C }\).
Edexcel C34 2019 June Q6
7 marks Standard +0.3
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-14_988_1120_123_395} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of the graph with equation \(y = | 4 x + 10 a |\), where \(a\) is a positive constant. The graph cuts the \(y\)-axis at the point \(P\) and meets the \(x\)-axis at the point \(Q\) as shown.
    1. State the coordinates of \(P\).
    2. State the coordinates of \(Q\).
  2. A copy of Figure 1 is shown on page 15. On this copy, sketch the graph with equation $$y = | x | - a$$ Show on the sketch the coordinates of each point where your graph cuts or meets the coordinate axes.
  3. Hence, or otherwise, solve the equation $$| 4 x + 10 a | = | x | - a$$ giving your answers in terms of \(a\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-15_860_1128_447_392} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} \(\_\_\_\_\) 7
Edexcel C34 2019 June Q7
9 marks Standard +0.3
7.
  1. Express \(5 \cos \theta - 3 \sin \theta\) in the form \(R \cos ( \theta + \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the exact value of \(R\) and give the value of \(\alpha\), in radians, to 4 decimal places. The height of sea water, \(H\) metres, on a harbour wall is modelled by the equation $$H = 6 + 2.5 \cos \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \sin \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
  2. Calculate the times at which the model predicts that the height of sea water on the harbour wall will be 4.6 metres. Give your answers to the nearest minute. \includegraphics[max width=\textwidth, alt={}, center]{a9870c94-0910-46ec-a54a-44a431cb324e-18_2257_54_314_1977}
Edexcel C34 2019 June Q8
11 marks Standard +0.3
8. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a9870c94-0910-46ec-a54a-44a431cb324e-22_524_1443_260_246} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \in \mathbb { R }$$
  1. Find \(\mathrm { f } ^ { \prime } ( x )\), writing your answer as a single fraction in its simplest form. The curve has two turning points, a maximum at point \(A\) and a minimum at point \(B\), as shown in Figure 2.
  2. Using part (a), find the coordinates of point \(A\) and the coordinates of point \(B\).
  3. State the coordinates of the maximum turning point of the function with equation $$y = \mathrm { f } ( 2 x ) + 4 \quad x \in \mathbb { R }$$
  4. Find the range of the function $$\operatorname { g } ( x ) = \frac { 6 x + 2 } { 3 x ^ { 2 } + 5 } , \quad x \leqslant 0$$
Edexcel C34 2019 June Q9
8 marks Standard +0.3
9.
  1. Using the formula for \(\sin ( A + B )\) and the relevant double angle formulae, find an
    identity for \(\sin 3 x\), giving your answer in the form $$\sin ( 3 x ) \equiv P \sin x + Q \sin ^ { 3 } x$$ where \(P\) and \(Q\) are constants to be determined.
  2. Hence, showing each step of your working, evaluate $$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 2 } } \sin 3 x \cos x d x$$ (Solutions based entirely on graphical or numerical methods are not acceptable.)
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Edexcel C34 2019 June Q10
9 marks Standard +0.3
  1. Use the binomial series to find the expansion of $$\frac { 1 } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), giving each term as a simplified fraction.
  2. Hence or otherwise, find the coefficient of \(x ^ { 2 }\) in the series expansion of
    1. \(\frac { 1 } { ( 2 + 6 x ) ^ { 3 } } \quad | x | < \frac { 1 } { 3 }\)
    2. \(\frac { 4 - x } { ( 2 + 3 x ) ^ { 3 } } \quad | x | < \frac { 2 } { 3 }\)